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Hysteresis and periodic solutions of semilinear and quasilinear wave equations. (English) Zbl 0658.35065

Recently much work has been done in the investigation of mathematical models of hysteresis phenomena and in applications in the theory of partial differential equations with hysteresis. The present paper is devoted to a detailed study (Sections 1, 2) of the Ishlinskiĭ hysteresis operator which has been introduced in the book of M. A. Krasnosel’skij and A. V. Pokrovskij [“Systems with hysteresis” (1983)]. In Sections 3 and 4 we prove the existence of \(\omega\)-periodic solutions to the problems \[ (1)\quad u_{tt}-u_{xx}\pm F(u)=g(t,x),\quad u(t,0)=u(t,\pi)=0, \]
\[ (2)\quad u_{tt}-F(u_ x)_ x=g(t,x),\quad u_ x(t,0)=u_ x(t,\pi)=0 \] for each (the assumptions are precised later) \(\omega\)-periodic right-hand side g, where F is the Ishlinskiĭ operator and \(\omega >0\) is arbitrary. These equations describe the forced longitudinal vibrations of a beam with elastico- plastic damping (1) and of an elastico-plastic beam (2). Equation (2) corresponds also to the one-dimensional Maxwell equation in a ferromagnetic medium. Notice that these existence results are due to special properties of the Ishlinskiĭ operator which enable us to obtain strong apriori estimates and to use the classical Galerkin method based on the compactness argument.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B10 Periodic solutions to PDEs
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35B45 A priori estimates in context of PDEs
35A35 Theoretical approximation in context of PDEs
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References:

[1] Krasnoselskiî, M.A., Pokrovskiî, A.V.: Systems with hysteresis. Moscow: Nauka 1983 (Russian)
[2] Visintin, A.: On the Preisach model for hysteresis. Nonlinear. Anal.8, 977–996 (1984) · Zbl 0563.35007 · doi:10.1016/0362-546X(84)90094-4
[3] Visintin, A.: Rheological models and hysteresis effects. Ist. Anal. Num. C.N.R., Pavia, Italy. Preprint no. 398 · Zbl 0633.73001
[4] Visintin, A.: Evolution problems with hysteresis in the source term. Ist. Anal. Num. C.N.R., Pavia, Italy. Preprint no. 326 · Zbl 0618.35053
[5] Besov, O.V., Il’in, V.P., Nikolskiî, S.M.: Integral representations of functions and embedding theorems (Russian). Moscow: Nauka 1975
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